729 



tangents Ihroiigh lluil point. From (his follows that the planes of 

 the V", louc'hing (f, envelop a cone of class ((n-^'2){n — J). 



Each conimon tangent plane of the two cones, contains a y", which 

 cuts / and touches (f ; for the number of those curves we tind there- 

 fore again {n-\-1){ii'' — 1). 



The two cones of class [n-^-IXn — 1), which are enveloped bj the 

 planes of the y", which touch two given planes have (?i-|-2)* {n — 1) 

 tangent planes in common. As many curves y" consequentli/ touch 

 tioo given planes. 



4. A surface ^"+^ belonging to the polar ray ƒ, contains a 

 number of y" with a node ; such a y" is the intersection of 2 with 

 a tangent plane passing through ƒ. 



In order to determine the number of those planes, we consider 

 the points which 2 outside ƒ, has in conimon with the polar surfaces 

 «" and /i" of two points A and B lying on ƒ. A plane ^ passing 

 through ƒ cuts these surfaces along two curves a"~^ and è""~^ which 

 cut ƒ in two groups of {n — 1) points At and Bk- If <p is made to 

 revolve round ƒ, these sets of {n — 1) points describe two projective 

 involutions so that a correspondence [n — l,?z — 1) arises on/. In 

 each coincidence C, ƒ is cut by two curves a"~"', l>"~^ lying in the 

 same plane (f ; there «" and /?" have therefore the same tangent 

 plane which contains at the same time the tangent of the curve 

 ff of the order {n^ — 1), which r«" and /i" have in common, apart from ƒ. 



The 2('/z — -1) points C are at the same time the coincidences of 

 the involution of the ?i*'' degree, which is determined on ƒ by the 

 curve y", out of which ^ is built up ; in each point C, ^ is there- 

 fore touched by the plane fp and moreover by the curve y. Conse- 

 quently Q has on ƒ 4(n — 1) points in common with S ; the number 

 of intersecting points of <fi and 2 lying outside /" amounts therefore 

 to (n'—l){7i-{-l) — Mn=zl) = {n—lYin-i-3). ') 



Through each polar ray ƒ pass consequently the planes of 

 {n — l)^(n-|-3) nodal curves y"o. 



The planes of the nodal curves y''^; envelop a cone of class 

 {n — l)-[n-{-S); the ])lanes of the ;/", which rest on a straight line /, 

 envelop a cone of class (ii-{-l). From this folloivs that the nodal 

 curves y",: form a surface A of order (^i+3) {n-\-'i) {n — 1)^ 



On a straight line ƒ lie n (n — 1)^ (n-|-3) points of the nodal curves 

 y"o-, of wdiich the planes pass through f; in the [)ole F the surface 

 A is cut by ƒ in (?z+3) {n — 1)' points. 



1) For w + 1 = 3, we duly (ind the live paii.s of lines wliich rest on a straight 

 line of a cubic surface. 



47* 



